= Solution
{c}
Differentiate the <worldline> with respect to $\theta$:
$$
\frac{dt}{d\theta}=\sqrt2T\cos\theta>0,\qquad \frac{dx}{d\theta}=-\sqrt2cT\kappa\sin\theta,\qquad\frac1c\frac{dx}{dt}=-\kappa\tan\theta.
$$
Since $|\tan\theta|\le1$ and $0<\kappa<1$, this is a future-directed <timelike curve>. The <proper time> element satisfies $c^2d\tau^2=c^2dt^2-dx^2$, hence
$$
\frac{d\tau}{d\theta}=\sqrt2T\sqrt{\cos^2\theta-\kappa^2\sin^2\theta}.
$$
Alice's total <proper time> is therefore
$$
\boxed{\tau_A=2\sqrt2T\int_0^{\pi/4}\sqrt{1-(1+\kappa^2)\sin^2\theta}\,d\theta.}
$$
This definite integral is an exact answer; it is an <elliptic integral of the second kind>. In notation $E(\alpha\mid m)=\int_0^\alpha\sqrt{1-m\sin^2u}\,du$, it is $2\sqrt2T E(\pi/4\mid1+\kappa^2)$. Although this parameter exceeds one, the integrand is real on the stated interval because its minimum squared value is $(1-\kappa^2)/2>0$.
The endpoint events have $(t,x)=(0,0)$ and $(2T,0)$. Sam stays at $x=0$ in his <inertial frame>, so his elapsed <proper time> is $2T$. For $\kappa>0$, the integrand is strictly less than $\cos\theta$ except at $\theta=0$, and $\tau_A<2T$.
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